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Multiplication of radical expressions is similar to multiplication of polynomials. Remember what a gas that was? Now we can experience that thrill ride all over again, and you don't even need to wait in an incredibly long line first.

When multiplying radical expressions, we give the answer in simplified form. Multiplying two monomial (1-term) radical expressions is the same thing as simplifying a radical term.

Sample Problem

Multiply .

We multiply the radicands to find .

Then, we simplify our answer to .

Sample Problem

Multiply

We distribute the and simplify the resulting terms:

Since these simplified terms have different radicands, there are no like terms to combine, so we're done. If you absolutely need a combining fix, we suggest experimenting with your little sister's poster paints. Hint: Yellow and blue make green.

To find the product of two binomial (2-term) radical expressions, we use FOIL. We're sure you've managed to preserve your memory of how to do this; good thing you wrapped it in FOIL.

Sample Problem

Multiply .

The first thing we do is simplify each radical term, if possible. We can replace with 2, and now the problem is

.

Now we use FOIL. First, outside, inside, last. It's all coming back to you now, right? Don't give us that look, everyone loves Celine.

Anyway, we start by multiplying the first terms:

,

which gives us

.

We multiply the outside terms:

,

which gives us

.

We multiply the inside terms:

which gives us

.

Finally, we multiply the last terms:

,

which gives us

.

Adding together all of our cute little products yields

.

Since the radicals have different radicands—one might even say "radically different radicands," which we will—this is as simplified as the answer gets. Rad.

Example 2

Example 3

The first thing we do, as always, is simplify each radical term if possible. The radical terms here are already simplified, so we can go straight to FOIL.

We multiply the first terms:

,

which gives us

.

We multiply the outside terms:

,

which gives us

.

We multiply the inside terms:

,

which gives us

.

We multiply the last terms:

,

which gives us

.

Finally, adding the products we got from each piece of FOIL gives us

.

We can't simplify this any, so we're done. Stick a fork in us.

Be Careful: When multiplying radicals, write neatly so you know what's part of the radicand and what isn't. This is no time for your usual hieroglyphic-doctor-signature-chicken-scratch.

is not the same thing as .

It's easy to make that mistake, so set yourself up for success. It's safer to write > as or or . However you write things, make sure you understand your own notation. If you forget or can't read what you wrote, no one will be able to help you. You'll be up a radical without a paddical.